发表机构
Indian Institute of Technology Delhi; Whitman College（德里印度理工学院; 惠特曼学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从第一性原理重新推导机械波动方程，证明其非不变性源于欧拉坐标表达，并提出拉格朗日导数形式与因式化欧拉形式两种重构，澄清伽利略变换局限及多普勒效应观测联系，兼具概念清晰性与教学价值。
AI 中文摘要
由牛顿定律导出的机械波动方程，在介质静止系中具有最简形式，但一旦用另一个惯性系的坐标表达，该形式似乎完全丧失——这引出了一个问题：对于任意惯性观测者，它能否以及如何被一致地写出。这一难题在本科教学中常被一带而过，通常通过将介质静止系视为“优先”系来回避，而未阐明这对作为物理定律基础的相对性原理意味着什么。我们从第一性原理重新审视波动方程，并证明这种表面上的非不变性源于其在欧拉坐标中的表达。我们提出两种重构形式：一种是在微观粒子力定律层面使用拉格朗日导数的显式不变形式，另一种是分离左右传播波动解的因式化欧拉形式。在此背景下，我们探讨了伽利略变换作为洛伦兹变换低速极限的局限性，并讨论了潜在的观测联系，特别是关于多普勒效应中波长变化的问题。我们的推导为中级和高级本科物理提供了概念清晰性和教学价值。
英文摘要
The mechanical wave equation, derived from Newton's laws, takes its simplest form in the rest frame of the medium, and appears to lose that form entirely once expressed in the coordinates of another inertial frame --- raising the question of whether, and how, it can be written consistently for an arbitrary inertial observer. This puzzle, often glossed over in undergraduate instruction, is typically set aside by invoking the rest frame of the medium as ``preferred,'' without clarifying what this means for the principle of relativity underlying physical law. We revisit the wave equation from first principles and demonstrate that this apparent non-invariance stems from its expression in Eulerian coordinates. We present two reformulations: a manifestly invariant form using Lagrangian derivatives applied at the level of microscopic particle force laws, and a factorized Eulerian form separating left and right-moving wave solutions. In this context, we explore limitations of Galilean transformation as the low-speed limit of Lorentz transformation, and discuss potential observational connections, particularly regarding wavelength changes in the Doppler effect. Our derivation provides both conceptual clarity and pedagogical value for intermediate and advanced undergraduate physics.
CommentsAccepted for publication in American Journal of Physics
